A patent-citation network grows one patent at a time
PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems
Many Parts, One Behaviour, Ch Growing networks: preferential attachment and scale-free degree distributions
Problem
A patent-citation network grows one patent at a time, time t counted in patents filed. Each new patent cites c=4 earlier patents, chosen with probability proportional to qi+a, where qi is the number of citations patent i has already received and a=8 is a fixed offset that gives even an uncited patent (qi=0) a nonzero chance of being cited. Let pq be the stationary fraction of patents with exactly q citations. (a) The total citation weight of all t existing patents at time t is ∑i(qi+a)=t(c+a), since the total number of citations made is ct. Using this, write the rate equation for Nq(t), substitute Nq=tpq, and show it reduces to pq=q+a+1+a/cq−1+apq−1,q≥1. (b) The telescoping product of this recursion is a ratio of gamma functions recognisable as a beta function, pq∝B(q+a,2+a/c), whose large-q behaviour B(x,y)∼Γ(y)x−y (quoted) gives a power-law tail pq∝q−γ with γ=2+a/c. Evaluate γ for this patent network. (c) Find the offset a′ (at the same c=4) that would instead reproduce the Barabási-Albert exponent γ=3, and comment on what the patent network's actual offset implies about its citation tail relative to a pure Barabási-Albert network.